STATISTICS IN TRANSITION new series, June 2020 Vol. 21, No. 2, pp. 173–187, DOI 10.21307/stattrans-2020-019 Submitted – 04.03.2019; accepted for publishing – 31.01.2020 Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process Piotr Szczepocki 1 ABSTRACT Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models. Key words: Ornstein–Uhlenbeck process, stochastic volatility, iterated filtering. 1. Introduction Barndorff-Nielsen and Shephard (2001) proposed a continuous-time stochastic volatility model (BN-S model), in which the logarithm of the asset price y t is assumed to be the solution of the following stochastic differential equation: dy t P EV t dt V t dB t (1) where B t tt is the Brownian motion, P R is the drift parameter, E R is the risk premium. Latent instantaneous volatility process V t tt is determined by the stochastic differential equation dV t OV t dt dz Ot (2) Department of Statistical Methods/Institute of Statistics and Demography/Faculty of Economics and Sociology/ University of Lodz, Poland. E-mail: [email protected]. ORCID: https://orcid.org/0000-0001-8377-3831. 174 P. Szczepocki: Application of iterated filtering… where O R and z Ot tt is pure jump Lévy process with stationary, independent and positive increments, and z . The process z Ot tt is called Background Driving Lévy Process (BDPL) of the process V t tt . Figure 1 presents examples of the pair of the processes z Ot tt and V t tt . There are several important features of such a stochastic volatility process defined by (2), some of which will be outlined in Section 2 on the basis of a series of Barndorff-Nielsen and Shephard papers (Barndorff-Nielsen and Shephard, 2001, 2002, 2003). A great number of estimation techniques have been proposed to estimate BN-S model. In their introductory paper (Barndorff-Nielsen and Shephard, 2001), Barndorff-Nielsen and Shephard employed a nonlinear least squares estimation and suggested other possible methods: Bayesian inference, quasi-likelihood inference by means of Kalman filter (for more details of Kalman filter implemented for BN-S model, see Szczepocki (2018)), estimation equations (Sørensen, 2000) and indirect estimation (Gourieroux, Monfort and Renault, 1993). In the following years much work on estimation was devoted to the Bayesian Markov Chain Monte Carlo (MCMC) approach: Roberts et al. (2004), Griffin and Steel (2006, 2010), Gander and Stephens (2007a,b), Frühwirth-Schnatter and Sögner (2009). Hubalek and Posedel (2006, 2011) proposed an estimator based on martingale estimating functions. Taufer, Leonenko and Bee (2011) introduced a characteristic function-based estimation method. Raknerud and Skare (2011) implemented an indirect inference method based on approximate Gaussian state space representation. Andrieu et al. (2010) proposed to use Particle Markov Chain Monte Carlo (PMCMC) estimation method, which combines particle filter with Bayesian inference. Chopin et al. (2013) proposed SMC2 algorithm, which substantially extends PMCMC. James et al. (2018) also used PMCMC for OU- Gamma Time Change version of BN-S model. In this paper we propose estimation based on iterated filtering. It is relatively a new class of methods for maximum likelihood inference introduced by Ionides et al. (2006) and substantially modified by Ionides et al. (2015). It is based on a series of filtering operations which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. In the discussion on (Andrieu et al., 2010) Anindya Bhadra (one of co-authors of Ionides et al., 2011) showed some results from applying the iterated filtering to a single example of BN-S model. However, he applied the initial version of iterated filtering (IF1) from Ionides et al. (2006). In this paper we employed the second generation version of iterated filtering (IF2) from Ionides et al. (2015). The paper is organized as follows. Section 2 presents background material on Barndorff-Nielsen and Shephard stochastic volatility model. Section 3 presents iterated filtering. Section 4 contains simulation results on estimation and Section 5 applications to real data. Section 6 gives concluding remarks. STATISTICS IN TRANSITION new series, June 2020 175 Figure 1. Two simulations of instantaneous volatility process with Gamma marginal (a) and (b), and corresponding Background Driving Lévy Process (c) and (d) Source: Own work using R software. 2. Barndorff-Nielsen and Shephard stochastic volatility model BN-S model has several important features which makes it very important for financial modelling. Firstly, instantaneous volatility V t tt moves up by jumps according to z Ot tt and tails off exponentially at the rate O . Thus, memory of the volatility process depends strictly on the rate O . High values of O result in high jumps, which are quickly discounted. On the contrary, a small value leads to a small jump but the process tails off slowly. Figure 1 shows the impact of O on the volatility process. Secondly, the time index of the process z Ot tt in (2) is chosen deliberately so that marginal distribution of V t does not depend on O . Barndorff-Nielsen and Shephard (2001) proved that for any one-dimensional self-decomposable distribution 176 P. Szczepocki: Application of iterated filtering… D there is a stationary Ornstein-Uhlenbeck process V t tt and Lévy process z Ot tt satisfying equation (2), for which marginal distribution of V t is D . The class of self-decomposable distribution includes many distributions important in financial econometrics: gamma, normal-inverse Gaussian, inverse Gaussian, tempered stable, variance gamma, symmetric gamma, the Euler’s gamma, Mexiner. (Schoutens, 2003) is a comprehensive reference text on financial application of self-decomposable distributions. Thirdly, although instantaneous volatility V t tt has discontinuous paths (due to jumps), integrated volatility t V t ³V u du (3) has continuous paths. Consequently, the resulting process of the logarithm of the asset price y t also has continuous paths. One advantage of stochastic volatility of Ornstein- Uhlenbeck type is that many important process characteristics are analytically tractable. For example, integrated volatility has a simple structure V t z Ot V t V (4) O Finally, the implication of the formula (1) is that log-returns observed at time n=1,..., T (we assume that time differences ' n tn tn are fixed and equal Δ) take the form: n' y dy t y n' y n ' (5) n ³ n ' and have conditional Normal distribution yn N P' EV n V n (6) where V n V n' V n ' . This discretely observed volatility V n (n=1,..., T) was called actual volatility by Barndorff-Nielsen and Shephard (2001). Marginal distribution of yn is a location scale mixture of normals. Thus, returns may capture empirical facts such as skewness and thick tails. Moreover, when ' o f marginal distribution of yn tends to normal distribution. Hence, non-normality of returns vanishes under temporal aggregation, which is another empirical fact observed in financial data. BN-S model has attracted much interest and research in mathematical finance and financial econometrics. Nicolato and Venardos (2003) studied equivalent martingale measures and provided closed-form prices for European call options for BN-S model. The minimal entropy martingale measure and numerical option pricing for BN-S STATISTICS IN TRANSITION new series, June 2020 177 model are investigated in (Benth and Karlsen, 2005) and (Benth and Meyer-Brandis, 2005). Hubalek and Sgarra (2009) provided option pricing by Esscher transform. Benth et al. (2003) considered Merton’s portfolio optimization problem in a Black and Scholes market with stochastic volatility of BN-S type. Benth et al. (2007) provided explicit evaluation of the variance swaps. Hubalek and Sgarra (2011) developed a semiexplicit valuation formula for geometric Asian options. 3. Iterated filtering 3.1. General remarks Iterated filtering (Ionides et al. 2006, 2015) are methods for maximum likelihood inference for state space models (SSMs). These models are also known as partially observed Markov Processes (POMP) or hidden Markov models (HMMs). SSMs consist of a pair of processes: X n Yn . The former is a Markov process (state process) which is not observed directly but may be estimated through the latter (observation processes). The observations of Yn are assumed to be conditionally independent given the X n (for details, see Durbin And Koopman, 2012). SSMs are very flexible and have been widely applied in economics, medicine, biology, mechanical system monitoring, patter recognition (see Chapter 1 in Cappé et al., 2008 for examples). However, estimation for SSMs is very challenging because likelihood functions are analytically intractable in general. Iterated filtering is one of the few if not the only available likelihood-based (based on the likelihood function for the full data), simulation-based (dynamics of the model is captured only via a simulator), frequentist (based on frequency interpretation of probability) methods for SSMs.